Selected topics from complex analysis

Selected topics from complex analysis

Blok 4A; April 23-June 29, 2007

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The Study Board in the Mathematical Sciences

If you want to find information about the mathematicians you hear about in the course, then look at MacTutor History of Mathematics Archive

The course is taught by Christian Berg . < berg@math.ku.dk> Information about the course will be given from this page.

The officiel desciption of the course Concerning evaluation we repeat from the official homepage: Assessment: Evaluation by three sets of problems each counting 1/3. Re-evaluation: Already passed sets of problems are not reevaluated but missing problem sets are replaced by a 30 min. oral exam.
The study board will collect evaluations of the course from the participants in the week June 2-10. Follow the link student evaluation
Schedule: Lectures: Monday 13.15-15.00 in Auditorium 10; Thursday 10.15-12.00 in Aud. 8 and 14.15-15.00 in Auditorium 7 First time Monday April 23.
Teaching material: J. K. Langley: "Postgraduate notes on complex analysis." This is a set of lecture notes available form the homepage of Prof. Langley at University of Nottingham. Langley's Notes
It will not be possible to cover all the material of the notes. We go through the first 4 chapters and then decide about additional chapters.
Problems to be solved.
  • First problem set: Handed out by May 7, deadline May 14 Problem1.pdf Remember to tell on the problem set if you want grades after the 13-scale or just pass/not passed
    Solutions to Problem set 1: solution1.pdf
  • Second problem set: Handed out by May 24, deadline June 4 problem2.pdf
    Solutions to Problem set 2: solution2.pdf
  • Third problem set: Handed out by June 11, deadline June 21 problem3.pdf
    Solutions to Problem set 3: solution3.pdf
  • Week 17: We will discuss the material of chapter 1. I started on Chapter 2 and defined the maximum modulus.
    Comments: On page 1, line 7 from below the same expression appears twice in the formula. One time is enough. On page 2 line 3 from below: In (i) the value L can also be - infinity. In the third line of the proof of Lemma 1.2.4 log s(Br) shall be a log^+
    The notions of density and logarithmic densities can be unified: Given a positive measure $\mu$ on the interval [0, infinity[, we can define the quotient $\mu(E\cap [0,r])/\mu([0,r])$ and take limsup, liminf as r tends to infinity, giving an upper and lower $\mu$ density of E. The cases treated corresponds to $\mu$ being Lebesgue measure and the measure $fd\mu$, where $f(x)=0$ for $0 \le x \le 1$ and $f(x)=1/x for x bigger than 1.
  • Week 18: I lectured on Chapter 2 including 2.2.4 on Monday. Thursday I will start on Lemma 2.2.5.
    I examined the possibility to move the lecture on Thursday to 13-14, but no auditoriums are available.
  • Week 19: This week there is only lecture on Monday 7, the lecture on Thursday 10 is cancelled, but you are supposed to work on the first problem set. This does not mean however that I cancel a lecture whenever you have problem sets!). I lectured about the beginning of Nevanlinna theory, and explained Poisson's formula, first for a holomorphic function in a neighbourhood of the set E. In this case it is actually a rather simple consequence of the residue theorem: Look at the second last line on page 15, assuming that U is holomorphic in a slightly larg er disc. The integrand has a simple pole at w=0 and no other poles inside the circle S(0,R). Therefore this integral is 2i\pi times the residue at w=0, but this is U(a), and this gives (3.1). If U has zeros and poles on S(0,R), we can apply the same technique to the contour Gamma_delta, and thus we get (3.2). Finally we let delta tend to zero and estimate that the contrubution from the small arcs tend to zero, and we also apply Lebesgue's theorem on dominated convergence.
    I told about the geometric interpretation of the Poisson kernel and the solution of the Dirichlet problem. Finally I proved the Poisson-Jensen formula (3.5) which follows from (3.1) by a trick: We multiply with some Möbius transformations, which remove that zeros and poles in the open disc and has the same absolute value on the circle S(0,R).
  • Week 20: I introduced the proximity function m(R,f) and the integrated counting function N(R,f), the sum of which is the Nevanlinna characteristic T(R,f), and we rewrote (3.6) as (3.13). The last thing I did was subsection 3.2.5.
  • Week 21: We start on 3.2.6 on Monday May 21. On May 21 I almost finished the proof of Lemma 3.3.1. On Thursday 24 I finished this Lemma and the next 2. The more technical lemmas 3.3.4 and 3.3.5 were used without proof to finish Henri Cartan's formula from 1929 about T(r,f). Henri Cartan, who is still alive (born 1904!), is son of Elie Cartan, famous for his work on Lie groups.
  • Week 22: Thursday May 31: I defined the order of a meromorphic function and proved that the set of meromorphic functions of order at most a is a sub-field of the field of meromorphic functions. This is not included in Langley's notes. In particular I proved that the order of a sum, difference, product and quotient of two meromorphic functions is less or equal to the maximum of the two orders with equality if the orders are different. Then I proved all the estimates of Section 3.4. I will start on 3.5 next week.
  • Week 23: Monday June 4: I handed out 3 handwritten pages about order of meromorphic functions and the formulations of Picard's great and little theorems. I lectured on Section 3.5 except the Examples 3.5.3 which I will cover on Thursday. I did the example and 3.6 on thursday.
  • Week 24: This week the lecture on MOnday June 11 will be replaced by a lecture Tuesday June 12, 13-15 in Aud. 10, but the Thursday lecture is unchanged. This is due to previous holidays.
    June 12: I lectured on 3.7 and finished page 34. I told a little about infinite products in general and handed out 6 pages from Rudin: Real and complex analysis, chapter 15. This is for self-study if you want to know more about infinite products than is covered by Langley's notes.
    June 14: I proved Theorem 3.7.5 which is called Borel's Theorem. I gave several examples of Borels theorem, related to the Gamma function and the sine function. I handed out handwritten notes, p. 1-5 about this. Then I continued to lecture about some general theorems about infinite products, following handwritten notes pages 1-6, also handed out. I finished pages 1-3. Misprints: Euler died 1783. Page 3, line 3 from bottom: sin(\varphi_0)=r/|z|.
  • Week 25: June 18: I finished the pages 4-6 of the notes: Some words about infinite products. On June 18 I handed out the following hand-written notes:Further on meromorphic and entire functions, pages 1-18. I had already lectured on the beginning of these notes on Monday, and went through the notes until page 16. I proved Mittag-Leffler's theorem about existance of a meromorphic function with prescribed poles and principal parts. I also proved a precise form of an entire function of finite order having a Picard exceptional value. The notes ends with an example of an entire function which tends to zero along any ray emanating from 0, but of course not being identical zero.
  • Week 26: Monday June 25:I finished the hand-written notes and lectured about a function H(z) of Hardy and Littlewood. It is closely related to the sine function, but the coefficient to z^{2k-1} in the power series for sin(z) is multiplied with zeta(2k) to get the power series for H(z). The function turns out to be undbounded on the real axis and the proof of that uses the distribution of the prime numbers. This ends the course.